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## G = C3×D5order 30 = 2·3·5

### Direct product of C3 and D5

Aliases: C3×D5, C5⋊C6, C152C2, SmallGroup(30,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C3×D5
 Chief series C1 — C5 — C15 — C3×D5
 Lower central C5 — C3×D5
 Upper central C1 — C3

Generators and relations for C3×D5
G = < a,b,c | a3=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Character table of C3×D5

 class 1 2 3A 3B 5A 5B 6A 6B 15A 15B 15C 15D size 1 5 1 1 2 2 5 5 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ4 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ5 1 -1 ζ32 ζ3 1 1 ζ6 ζ65 ζ32 ζ3 ζ32 ζ3 linear of order 6 ρ6 1 -1 ζ3 ζ32 1 1 ζ65 ζ6 ζ3 ζ32 ζ3 ζ32 linear of order 6 ρ7 2 0 2 2 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ8 2 0 2 2 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ9 2 0 -1-√-3 -1+√-3 -1-√5/2 -1+√5/2 0 0 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 complex faithful ρ10 2 0 -1-√-3 -1+√-3 -1+√5/2 -1-√5/2 0 0 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 complex faithful ρ11 2 0 -1+√-3 -1-√-3 -1+√5/2 -1-√5/2 0 0 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 complex faithful ρ12 2 0 -1+√-3 -1-√-3 -1-√5/2 -1+√5/2 0 0 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 complex faithful

Permutation representations of C3×D5
On 15 points - transitive group 15T3
Generators in S15
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)

G:=sub<Sym(15)| (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)>;

G:=Group( (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14) );

G=PermutationGroup([[(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14)]])

G:=TransitiveGroup(15,3);

Regular action on 30 points - transitive group 30T4
Generators in S30
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 23)(7 22)(8 21)(9 25)(10 24)(11 28)(12 27)(13 26)(14 30)(15 29)

G:=sub<Sym(30)| (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,20)(2,19)(3,18)(4,17)(5,16)(6,23)(7,22)(8,21)(9,25)(10,24)(11,28)(12,27)(13,26)(14,30)(15,29)>;

G:=Group( (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,20)(2,19)(3,18)(4,17)(5,16)(6,23)(7,22)(8,21)(9,25)(10,24)(11,28)(12,27)(13,26)(14,30)(15,29) );

G=PermutationGroup([[(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,23),(7,22),(8,21),(9,25),(10,24),(11,28),(12,27),(13,26),(14,30),(15,29)]])

G:=TransitiveGroup(30,4);

C3×D5 is a maximal subgroup of   C3⋊F5  C5⋊F7  D65⋊C3  F16⋊C2
C3×D5 is a maximal quotient of   C5⋊F7  D65⋊C3  F16⋊C2

Polynomial with Galois group C3×D5 over ℚ
actionf(x)Disc(f)
15T3x15+2x14+2x13-2x11-12x10+37x9-2x8+37x7+37x6-39x5-47x4-22x3-6x2+1710·476·836·22132·23512

Matrix representation of C3×D5 in GL2(𝔽19) generated by

 11 0 0 11
,
 18 5 14 5
,
 5 18 5 14
G:=sub<GL(2,GF(19))| [11,0,0,11],[18,14,5,5],[5,5,18,14] >;

C3×D5 in GAP, Magma, Sage, TeX

C_3\times D_5
% in TeX

G:=Group("C3xD5");
// GroupNames label

G:=SmallGroup(30,2);
// by ID

G=gap.SmallGroup(30,2);
# by ID

G:=PCGroup([3,-2,-3,-5,218]);
// Polycyclic

G:=Group<a,b,c|a^3=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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